Class number behavior in a two-tiered tower of ${\mathbb{Z}}_p$-extensions

Abstract

Let $k_{\infty }$ be the cyclotomic ${\mathbb{Z}}_p$-extension field of an algebraic number field $k$. Moreover, we take a ${\mathbb{Z}}_p$-extension $K_{\infty }$ over $k_{\infty }$. In this paper, we study the behavior of the $p$-part of the class number of certain intermediate fields of $K_{\infty }/k$. We also consider the structure of the unramified Iwasawa module of $K_{\infty }/k_{\infty }$ for several cases.

15 pages. Several places were modified. References were added. Some contents were added or updated